COMPUTATIONAL MODELLING OF ADIRECTIONAL SOLIDIFICATIONPROCESSA. J. Williams, G. A. Taylor, T. N. Croft,M. Cross and P. D. Brown�

Centre for Numeri al Modelling and Pro ess Analysis,S hool of Computation and Mathemati al S ien es,University of Greenwi h,London SE10 9LSemail: A.J.Williams�gre.a .uk� Rolls-Roy e pl , Elton Road, PO Box 31,Derby DE24 8BJemail: paul.brown�rolls-roy e. om

Aims of the Resear h Proje t� To generate a omputational mesh that represents an ide-alised geometry for the investment asting of a turbine blade.The assembly onsists of the investment shell and ore, thealloy omponent and the hill.� To develop a moving boundary ondition whi h models thewithdrawal of the assembly from the furna e, thus ontrollingthe dire tion and speed of solidi� ation.� To perform a fully oupled thermo-me hani al analysis ofthe withdrawal of the assembly from the furna e.Approa h� An idealised symmetri geometry has been assumed to re-du e the omputational e�ort required to perform the anal-ysis.� The thermo-elasti analysis has been modelled using a 3D ontinuum approa h.� Finite volume te hniques have been used to dis retize thegoverning equations.

Investment Casting� Investment asting is used to produ e omponents for use inair raft, ele troni , opti al and medi al equipment.� The pro ess enables manufa turers to produ e omplex om-ponents with a high degree of pre ision.� The omponents have ex ellent surfa e quality.Dire tional Solidi� ation� Allows the manufa turer to ontrol the speed of solidi� ationand govern growth of solid grain in the material.� Dire tional solidi� ation improves the quality and stru turalintegrity of the omponent thus in reasing the life y le andreliability of the produ t.GeometryThe geometry onsists of four se tions:� erami ore� erami shell� metal alloy� opper hillThe top and bottom of the ore are atta hed to the shell, andthe hill is in pla e at all times.

XY

Z

� Cerami shell� Alloy-Cerami ore� Copper hill ?

60.288 m

Figure 1: A verti al ross-se tion of the geometry

X

Y

Z

� -0.076 mFigure 2: A horizontal ross-se tion through main body of geometry

MeshThe symmetri mesh is made up of 8-noded hexahedral elements.

XY

Z

No. of elements = 45156No. of nodes = 53647Initial ConditionsThe temperatures of the shell, alloy and ore are set initially to1773K. The hill is given an initial temperature of 293K.Material PropertiesTemperature dependent material properties are used for the thermo-elasti analysis.

Governing EquationsTo model the thermo-elasti behaviour of the pro ess the follow-ing equations must be solved in a oupled fashion.The s alar heat equation with phase hange is�(� PT )�t = div(kgrad T )� Lh�(�f)�twhere� - density P - spe i� heat apa ityk - thermal ondu tivity f - liquid fra tionLh - latent heat of solidi� ationThe equilibrium equations des ribing the solid me hani s be-haviour are, in matrix form,[L℄Tf��g = 0where [L℄ is the di�erential operator and f��g is the Cau hystress in rement.The stress is related to elasti strain su h thatf��g = [D℄f��eg = [D℄ (f��g � f��thg)where [D℄ is the elasti ity matrix, and f��eg, f��g and f��thgare the elasti , total and thermal in remental strain, respe tively.The thermal strain is related to the in remental temperature hange, �T , as followsf��thg = f��T; ��T; ��T; 0; 0; 0gTwhere � is the oeÆ ient of thermal expansion.

Thermal and Me hani al Boundary ConditionsTo model the withdrawal from the furna e the geometry is �xedin spa e and a moving radiative boundary ondition is appliedto the external surfa e of the shell.Initially the ambient temperature TA satis�es the following tem-perature pro�le, where the top of the hill is lo ated at z = 0m.

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

200 400 600 800 1000 1200 1400 1600 1800TA (K)z (m)

Figure 3: Ambient temperature pro�leTo simulate the withdrawal the ambient temperature pro�le movesupwards at a onstant rate.Newton ooling boundary onditions are applied to the baseof the shell and hill, where the heat transfer oeÆ ients are10W/mK and 6000W/mK repe tively. An adiabati ondition isapplied to the top of the geometry and the symmetry plane.Any gap whi h o urs between the shell and alloy as the alloy ools and ontra ts is treated as a va uum.Fixed displa ement boundary onditions are applied to inhibitpenetration of the shell or ore by the alloy.

1a) 1b)

2a) 2b)

3a) 3b)

293416540663786910.103E4.116E4.128E4.14E4.153E4.165E4

.909E-1

.182

.273

.364

.455

.545

.636

.727

.818

.909

293416540663786910.103E4.116E4.128E4.14E4.153E4.165E4

.909E-1

.182

.273

.364

.455

.545

.636

.727

.818

.909

293416540663786910.103E4.116E4.128E4.14E4.153E4.165E4

0000000000

Figure 4: The ontours of a) temperature and b) liquid fra tion at 1)t = 30, 2) t = 60 and 3) t = 120 mins.

1a) 1b)

2a) 2b)

3a) 3b)

.2E9

.364E9

.527E9

.691E9

.855E9

.102E10

.118E10

.135E10

.151E10

.167E10

.184E10

.2E10

.2E9

.364E9

.527E9

.691E9

.855E9

.102E10

.118E10

.135E10

.151E10

.167E10

.184E10

.2E10

.2E9

.364E9

.527E9

.691E9

.855E9

.102E10

.118E10

.135E10

.151E10

.167E10

.184E10

.2E10

.2E9

.364E9

.527E9

.691E9

.855E9

.102E10

.118E10

.135E10

.151E10

.167E10

.184E10

.2E10

.2E9

.364E9

.527E9

.691E9

.855E9

.102E10

.118E10

.135E10

.151E10

.167E10

.184E10

.2E10

.2E9

.364E9

.527E9

.691E9

.855E9

.102E10

.118E10

.135E10

.151E10

.167E10

.184E10

.2E10

Figure 5: The e�e tive stress ontours on a) whole geometry and b) thealloy with the deformation magni�ed (�10) at 1) t = 30, 2) t = 60 and3) t = 120 mins.